Class 9 Maths Chapter 3 Coordinate Geometry – Exercise 3.3 NCERT Solutions

Introduction

Exercise 3.3 introduces the distance formula in coordinate geometry. You’ll learn how to calculate the distance between two points, verify collinearity, and check properties of triangles and quadrilaterals.

Key Formula Used

• Distance Formula:

$$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$$

Solved Questions (Step by Step)

Q1. Find the distance between points $(2,3)$ and $(4,1)$.

Solution:

$$d=\sqrt{(4-2{)}^2+(1-3{)}^2}=\sqrt{2^2+(-2{)}^2}=\sqrt{4+4}=\sqrt{8}=2\sqrt{2}$$

Q2. Find the distance between $(-1,-2)$ and $(3,2)$.

Solution:

$$d=\sqrt{(3-(-1){)}^2+(2-(-2){)}^2}=\sqrt{4^2+4^2}=\sqrt{16+16}=\sqrt{32}=4\sqrt{2}$$

Q3. Find the distance between $(0,0)$ and $(5,12)$.

Solution:

$$d=\sqrt{(5-0{)}^2+(12-0{)}^2}=\sqrt{25+144}=\sqrt{169}=13$$

Q4. Find the distance between $(7,-3)$ and $(7,5)$.

Solution:

$$d=\sqrt{(7-7{)}^2+(5-(-3){)}^2}=\sqrt{0+8^2}=8$$

Q5. Find the distance between $(-2,4)$ and $(3,4)$.

Solution:

$$d=\sqrt{(3-(-2){)}^2+(4-4{)}^2}=\sqrt{5^2+0}=5$$

Q6. Verify if points $(1,1),(4,1),(7,1)$ are collinear.

Solution:

• Distance between $(1,1)$ and $(4,1)$ = 3.

• Distance between $(4,1)$ and $(7,1)$ = 3.

• Distance between $(1,1)$ and $(7,1)$ = 6.

• Since 3 + 3 = 6, points are collinear.

Q7. Verify if points $(0,0),(2,2),(4,4)$ are collinear.

Solution:

• Distance between $(0,0)$ and $(2,2)$ = $\sqrt{8}=2\sqrt{2}$.

• Distance between $(2,2)$ and $(4,4)$ = $\sqrt{8}=2\sqrt{2}$.

• Distance between $(0,0)$ and $(4,4)$ = $\sqrt{32}=4\sqrt{2}$.

• Since $2\sqrt{2}+2\sqrt{2}=4\sqrt{2}$, points are collinear.

Q8. Find the distance between $(3,-2)$ and $(-3,-2)$.

Solution:

$$d=\sqrt{(-3-3{)}^2+(-2-(-2){)}^2}=\sqrt{(-6{)}^2+0}=6$$

Q9. Find the distance between $(0,5)$ and $(0,-5)$.

Solution:

$$d=\sqrt{(0-0{)}^2+(-5-5{)}^2}=\sqrt{0+(-10{)}^2}=10$$

Q10. Find the distance between $(2,2)$ and $(5,6)$.

Solution:

$$d=\sqrt{(5-2{)}^2+(6-2{)}^2}=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5$$

Q11. Find the distance between $(-4,-3)$ and $(0,0)$.

Solution:

$$d=\sqrt{(0-(-4){)}^2+(0-(-3){)}^2}=\sqrt{4^2+3^2}=\sqrt{16+9}=\sqrt{25}=5$$

Q12. Find the distance between $(1,-1)$ and $(-1,1)$.

Solution:

$$d=\sqrt{(-1-1{)}^2+(1-(-1){)}^2}=\sqrt{(-2{)}^2+(2{)}^2}=\sqrt{4+4}=\sqrt{8}=2\sqrt{2}$$

Q13. Find the distance between $(2,3)$ and $(2,-3)$.

Solution:

$$d=\sqrt{(2-2{)}^2+(-3-3{)}^2}=\sqrt{0+(-6{)}^2}=6$$

Q14. Find the distance between $(5,0)$ and $(-5,0)$.

Solution:

$$d=\sqrt{(-5-5{)}^2+(0-0{)}^2}=\sqrt{(-10{)}^2}=10$$

Q15. Find the distance between $(7,1)$ and $(1,7)$.

Solution:

$$d=\sqrt{(1-7{)}^2+(7-1{)}^2}=\sqrt{(-6{)}^2+6^2}=\sqrt{36+36}=\sqrt{72}=6\sqrt{2}$$

FAQs (10 for Exercise 3.3)

1. Q: What is the distance formula used for? A: To calculate the distance between two points in the plane.

2. Q: How do you check collinearity using distance formula? A: If sum of two distances equals the third, points are collinear.

3. Q: Can the distance formula give negative values? A: No, distance is always non‑negative.

4. Q: What is the distance between a point and itself? A: Zero.

5. Q: How do you find the distance between points on x‑axis? A: Subtract their x‑coordinates.

6. Q: How do you find the distance between points on y‑axis? A: Subtract their y‑coordinates.

7. Q: Why is the distance formula important? A: It helps verify geometric properties like collinearity and side lengths.

8. Q: What is the distance between $(0,0)$ and $(a,b)$? A: $\sqrt{a^2+b^2}$.

9. Q: Can the distance formula be extended to 3D? A: Yes, by adding the z‑coordinate term.

10. Q: What is the shortest distance between two points? A: The straight line joining them, calculated by the distance formula.

Solved Questions (Step by Step)

We solved 15 representative problems earlier. Here’s the consolidated set:

1. Distance between $(2,3)$ and $(4,1)$ = $2\sqrt{2}$.

2. Distance between $(-1,-2)$ and $(3,2)$ = $4\sqrt{2}$.

3. Distance between $(0,0)$ and $(5,12)$ = 13.

4. Distance between $(7,-3)$ and $(7,5)$ = 8.

5. Distance between $(-2,4)$ and $(3,4)$ = 5.

6. Points $(1,1),(4,1),(7,1)$ are collinear.

7. Points $(0,0),(2,2),(4,4)$ are collinear.

8. Distance between $(3,-2)$ and $(-3,-2)$ = 6.

9. Distance between $(0,5)$ and $(0,-5)$ = 10.

10. Distance between $(2,2)$ and $(5,6)$ = 5.

11. Distance between $(-4,-3)$ and $(0,0)$ = 5.

12. Distance between $(1,-1)$ and $(-1,1)$ = $2\sqrt{2}$.

13. Distance between $(2,3)$ and $(2,-3)$ = 6.

14. Distance between $(5,0)$ and $(-5,0)$ = 10.

15. Distance between $(7,1)$ and $(1,7)$ = $6\sqrt{2}$.

FAQs (10 for Exercise 3.3)

1. Q: What is the distance formula used for? A: To calculate the straight‑line distance between two points in the plane.

2. Q: How do you check collinearity using distance formula? A: If sum of two distances equals the third, points are collinear.

3. Q: Can the distance formula give negative values? A: No, distance is always non‑negative.

4. Q: What is the distance between a point and itself? A: Zero.

5. Q: How do you find the distance between points on x‑axis? A: Subtract their x‑coordinates.

6. Q: How do you find the distance between points on y‑axis? A: Subtract their y‑coordinates.

7. Q: Why is the distance formula important? A: It helps verify geometric properties like collinearity and side lengths.

8. Q: What is the distance between $(0,0)$ and $(a,b)$? A: $\sqrt{a^2+b^2}$.

9. Q: Can the distance formula be extended to 3D? A: Yes, by adding the z‑coordinate term.

10. Q: What is the shortest distance between two points? A: The straight line joining them, calculated by the distance formula.

Conclusion

Exercise 3.3 has 15 questions that strengthen your ability to use the distance formula, check collinearity, and verify

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